Materials Transactions, Vol. 47, No. 11 (2006) pp. 2682 to 2689
Special Issue on Advances in Computational Materials Science and Engineering IV
#2006 The Japan Institute of Metals
Grain Boundary Decohesion by Sulfur Segregation in Ferromagnetic Iron
and Nickel —A First-Principles Study—
Masatake Yamaguchi, Motoyuki Shiga and Hideo Kaburaki
Center for Computational Science and e-Systems (CCSE), Japan Atomic Energy Agency (JAEA),
Tokai-mura, Ibaraki 319-1195, Japan
Using first-principles calculations, we simulate grain boundary decohesion (embrittlement) in ferromagnetic bcc Fe3(111)[11 0] and fcc
Ni5(012)[100] symmetrical tilt grain boundaries by progressively adding sulfur atoms to the boundaries. We calculate the segregation energy
of sulfur atom, tensile strength, and cohesive energy of the grain boundaries. We show that a certain amount of sulfur segregation (two atomic
layers, 14.4 atom/nm2 ) is energetically possible to realize considering the calculated segregation energies. At this concentration, the tensile
strength and the cohesive energy of the grain boundaries reduce by one order of magnitude comparing with no segregation case.
[doi:10.2320/matertrans.47.2682]
(Received May 22, 2006; Accepted August 9, 2006; Published November 15, 2006)
Keywords: grain boundary, embrittlement, first-principles calculation, sulfur, iron, nickel
1.
Introduction
It is well known that a small amount of impurity (solute)
elements bring about a drastic change on the mechanical
properties of metals. For example, sulfur segregates to grain
boundaries (GBs) of metals and makes metals brittle.
However, the microscopic mechanism of this grain boundary
decohesion (embrittlement) has been not known in detail.
Recently, we have modeled the grain boundary decohesion
(embrittlement) by progressively adding sulfur atoms to a
nickel grain boundary.1–3) Our first-principles calculations
have revealed that the weakening of the boundary is caused
by the aggregation of sulfur atoms at the boundary, which
repel each other. Because the nickel-sulfur bonds are stronger
than the sulfur-sulfur bonds, the sulfur atoms are forced into
non-ideal bonding and therefore push apart each other when
they are neighboring.
In this paper, we show a refined result for the Ni-S system
and add a new result for the Fe-S system. The improvement
points in the method of calculations are as follows; (i) spinpolarized calculations are performed, (ii) unit cell including
vacuum region is used; this cell allows the grain boundary
sliding, (iii) the segregation energies for larger number of
segregation configurations of sulfur atoms are calculated, (iv)
structure relaxation is introduced into tensile test calculations, and the cohesive energy of grain boundary is also
shown with tensile strength. Although the results for fcc
Ni5 GB do not change significantly from our previous
results,1–3) the plausibility of these results are more firmly
confirmed by the above improvements.
2.
Methfessel-Paxton smearing method with 0.1-eV width is
used. All calculations are done for ferromagnetic state.
2.2 Modeling of grain boundary
It is known that the grain boundary embrittlement is caused
by the weakening of large angle (more than 15 ) grain
boundaries. In the large angle grain boundaries, we choose
bcc Fe3(111)[11 0] and fcc Ni5(012)[100] symmetrical
tilt grain boundaries (STGB) for the following reasons.
Table 1 shows some characters of the GBs. Although there
is a difference in crystal structure (bcc vs. fcc), tilt angles of
these two GBs are large and similar with each other. In
addition, the atomic areal density of atoms in each atomic
plane parallel to the GB plane is almost the same for the two
GBs (7.2 atom/nm2 ).
Figure 1 shows the GB structures. The atomic sites are
labeled by numbers and characters. The site 0 indicates grain
boundary vacancy site, in which S atom can enter. The other
sites are Fe sites in Figs. 1(a)–(b) or Ni sites in Figs. 1(c)–(d),
at which S atom can be substituted for Fe or Ni atom.
It is well known that the disordering of atomic positions at
the GB almost disappears in a few atomic layers (about
0.5 nm). For this reason, the unit cells shown in Fig. 1 have
sufficient length (2.55 nm) in c-axis direction. As for the
lateral direction (in ab plane), these cells have 4 (2 2) metal
atoms (7.2 atom/nm2 ) in each atomic plane, where atoms are
on crystallographic equivalent sites.
These unit cells have vacuum regions. The distance
between the two surfaces (upper and lower surfaces) that
sandwich the vacuum region is about 1 nm, which is so long
Method of Calculation
2.1 Electronic structure calculation
The electronic structure calculations and crystal structure
relaxations by force minimizations are performed using
Vienna Ab initio Simulation Package (VASP) with Projector
Augmented Wave (PAW) potentials.4–7) The cutoff energy
for the plane wave basis set is 280 eV. The Monkhorst Pack
k-point mesh is 3 3 1 for both Fe3 and Ni5 GBs. The
Table 1 Grain boundary characters (hkl) indicates the crystallographic
plane parallel to the GB, and [hkl] the tilt axis. The area of (hkl) plane in
the unit cell is shown.
bcc Fe3(111)[11 0]
fcc Ni5(012)[100]
tilt angle
70.5
53.1
Area of (hkl) plane
0.556 nm2
0.554 nm2
atomic density
7.2 atom/nm2
7.2 atom/nm2
Grain Boundary Decohesion by Sulfur Segregation in Ferromagnetic Iron and Nickel —A First-Principles Study—
(a)
(b)
(c)
(d)
c=<012>
c=<111>
-11
<101>
1a
1b
1a
-4
-3
-2
-3
2
3
GB
-2
1
1 0
1 0
-12
-11
-10
-7
2
2a
0a
3
2b
0b
1d
0c
1a
-6
-3
1
1a
3
1b
1b
1a 2a
1c
-
1a
<100>
2d
2c 0d
1c
4
-3
-2
1
0
2
3
-2
0
2
GB
1
0c
1c
<111>=c
11
-
a=<101>
2d
0d
1d
2c
0b
1c
2b
0a
1a
1b
1a
- -
6
<121>=b
<110>=b
7
10
2683
11
<012>=c
12
a=<100>
Fig. 1 Unit cell modeling of bcc Fe3(111)[11 0] and fcc Ni5(012)[100] symmetrical tilt grain boundary (STGB). Unit cell shapes are
shown by solid lines. Atomic sites (layers) are indicated by numbers (2, 0, 1, 2, etc.), in which the lateral atomic sites are indicated by
characters (a, b, c, d). The radius of sphere except for site 0 is 0.1 nm. (a) Side view of the bcc Fe3 STGB unit cell. (b) Top view of the
fractured free surface at the bcc Fe3 STGB. (c) Side view of the fcc Ni5 STGB unit cell. (d) Top view of the fractured free surface at
the fcc Ni5 STGB.
(a)
(b)
FS
(c)
confirmed that this energy is almost the same as the total
energy of one metal atom calculated using a primitive unit
cell (1 atom/u.c.). The free surface energy is obtained by
FS;0
FS;0
Bulk;Ni
NNi
(NNi
: number of Ni atoms in the
subtracting Etot;atom
FS cell) from the total energy of the FS cell.
FS
2.3 Calculations of binding energies
Using these unit cells, we calculate the binding energy of
segregated S atoms (EbGB ðconfig.Þ) depending on segregated
sulfur configuration in the following way.
GB
GB
atom;S
EbGB ðconfig.Þ ¼ Etot
ðNNi ; NS Þ Ns Etot
FS
FS
GB cell
GB;0
GB;0 Bulk;Ni
GB
Etot
ðNNi
; 0Þ ðNNi NNi
ÞEtot;atom
FS cell
Bulk cell
Fig. 2 (a) This is the same as Fig. 1(a). We refer this cell as ‘‘GB cell’’.
(b) This cell does not have grain boundary (GB) but has two free surfaces
(FSs) and the same number (76) of Fe atoms in similar to the GB cell. We
refer this cell as ‘‘FS cell’’. (c) This cell has the same a and b-axis lengths
of GB, FS cells. The c-axis length of this ‘‘Bulk cell’’ is 2.45(2.36) nm for
Fe(Ni) system, which is slightly shorter than that (2.55 nm) of GB, FS cells
to include bulk crystals of 120 Fe(Ni) atoms.
that the interaction between the two surfaces is negligible.
This vacuum region is introduced to allow GB sliding; the
upper half metal block can move freely parallel to the GB
plane with respect to the lower half block.
The following explanation in Ni case holds also in Fe case.
From now on, we refer this unit cell as ‘‘GB cell’’. On the
other hand, we also use a different cell that does not include
grain boundary, which is referred as ‘‘FS cell’’ as shown in
Fig. 2(b). The total energy difference between the two cells
corresponds to grain boundary energy. The total energy of
Bulk;Ni
bulk system per one metal atom, Etot;atom
, is calculated using
Bulk cell (120 atom/u.c.) as shown in Fig. 2(c). We
Here, this binding energy is calculated with respect to the
atom;S
total energy of an isolated sulfur atom in vacuum, Etot
, and
Bulk;Ni
also to that of metal (Ni) bulk crystal per one atom, Etot;atom
.
GB
Etot
ðNNi ; NS Þ is the calculated total energy for the relaxed
structure of the system that includes NNi Ni atoms and NS S
atom;S
atoms using the GB unit cell. Etot
is the calculated total
energy of a S atom that exists in vacuum (1 1 1 in nm).
GB;0
NNi
is the number of Ni atoms without S segregation in the
GB cell.
The binding energy of one S atom in fcc bulk Ni crystal is
calculated as follows.
FS;0
FS
EbBulk ¼ Etot
ðNNi
1; 1Þ
FS;0
atom;S
FS
Bulk;Ni
Etot
ðNNi
; 0Þ þ Etot;atom
Etot
FS;0
FS
Here, Etot
ðNNi
1; 1Þ is calculated when one S atom is
substituted for one Ni atom near the center of the FS cell.
This corresponds to the situation that one S atom exists in
inner bulk region, and thereby give an energy of solution.
Using the above energies, the total segregation energy for
S atoms is calculated as follows.
total
Eseg
ðconfig.Þ ¼ EbGB ðconfig.Þ Ns EbBulk
2684
M. Yamaguchi, M. Shiga and H. Kaburaki
40
0.6
0.4
773 (K), 25 (at.ppm)
773 (K), 50 (at.ppm)
773 (K), 100 (at.ppm)
773 (K), 200 (at.ppm)
1073 (K), 25 (at.ppm)
1073 (K), 50 (at.ppm)
1073 (K), 100 (at.ppm)
1073 (K), 200 (at.ppm)
0.2
0.5
1.0
1.5
5
6
7
8
9 10 11
3
fixed grip
20
4
10
0
2
fixed grip
2
1
3
-2
1
2.0
Segregation energy, –Eseg / eV atom-1
Fig. 3 McLean’s curves. For each curve, ageing temperature T(K) and
bulk impurity concentration Cbulk (atomic perts per million, atppm) are
shown.
This energy indicates how much energy is obtained by
moving S atoms from inner bulk region to the GB region. The
av
average segregation energy per one sulfur atom, Eseg
ðconfig.Þ
total
is obtained to divide Eseg ðconfig.Þ by Ns . Since the total
energy of the system is negative, the larger the negative
segregation energy, the more S atom can segregate.
Figure 3 shows some plots of McLean’s equation8) for
some segregation conditions (ageing temperature and bulk
impurity concentration). McLean’s equation is based on a
simple statistical model to predict segregation concentration
after the system reaches thermal equilibrium. McLean
derived the temperature dependence of grain boundary
segregation to be,
CGB ¼
4
30
Cohesive energy, 2γ / J m
fracture plane
0.8
0.0
0.0
5
Calculated total energy
f(x)
df(x)/dx
Tensile stress, σ / GPa
Occupation at segregation site, CGB
1.0
Cbulk expðEseg =RTÞ
;
1 þ Cbulk expðEseg =RTÞ
where CGB is the grain boundary segregation concentration
(occupation: 0–1), Cbulk the bulk impurity concentration
(0–1), Eseg the segregation energy at segregation site in grain
boundary region. Here, the concentration dependence of Eseg
is not taken into account.
From Fig. 3, we can understand that the value of 1:0
eV/atom is so large segregation energy that the segregation
concentration (occupation) can be more than 0.5 (50%) even
for low bulk concentration (25 atppm) and high ageing
temperature (1073 K).
2.4 Tensile test calculations
Here, we show how to calculate the cohesive energy (2)
and the maximum tensile stress (max , tensile strength). In our
previous paper,1–3) we performed a rigid-type tensile test
calculation, where we set a fracture plane and then the upper
and lower half crystal blocks are gradually separated without
structure relaxations. This calculation has a large numerical
error especially for high segregation concentration cases,
since the sulfur atoms can be much more stable on the
fractured free surfaces than in the GB as stated later. In this
work, we introduce the structure relaxation in the tensile test
calculation.
Ideally, the tensile test calculations should be done by
0
0.0
0
0.1
0.2
0.3
0.4
Separation / nm
0.5
0.6
Fig. 4 Method of fitting in tensile test calculations. This is an example for
clean bcc Fe3 GB.
repeating the following process: adding a small strain to the
system and subsequently making structure relaxations and
optimizing lateral lattice parameters (considering Poisson’s
ratio). However, this ideal method is very time-consuming.
Furthermore, the structure relaxation becomes difficult to
converge when the system has a large strain. For these
reasons, we take an approximate method as stated below.
Figure 4 shows an example of tensile test calculations for
clean bcc Fe3 GB. We take the following steps.
(i) A fracture plane parallel to the GB plane is assumed
(point 0).
(ii) The upper and lower crystal blocks at the fracture plane
are rigidly separated by some distances (0.05 nm, 0.1 nm,
0.15 nm, . . . 0.5 nm, etc.). About 10 sampling points that have
different separation distances are prepared.
(iii) For each sampling point, structure relaxation is performed. Here, a few atomic layers close to the free surface are
fixed to keep the two crystal blocks separated. This is usually
called ‘‘fixed grip’’ method as shown in Fig. 4. After the
structure relaxation, about 10 total energies depending on the
separation distances are obtained.
(iv) The cohesive energy (2) is easily obtained from the
difference between the two total energies; one is the energy
of the GB without separation (point 0), and the other is the
energy when the separation distance is so large that the
energy does not change any more (point 10 or 11).
(v) The maximum tensile stress is obtained as follows.
A simple function,
x
x
f ðxÞ ¼ 2 2 1 þ
exp ;
is fitted to the calculated total energies. This function is
well known as a universal binding curve, proposed by Rose
et al.,9) that describes the bonding nature between atoms
well. When fitting, we ignore some total energy points where
the separation distance is small (point 1–4 in Fig. 4). In this
small separation region, the total energies depend on the
lateral lattice parameters (a, b) due to Poisson’s ratio, and on
how many atomic layers are relaxed near the fracture plane.
In addition, it is difficult to converge because strain energy is
stored. For these difficulties, we do not use point 1–4 and
instead use point 5–11 for fitting. In this large separation
Grain Boundary Decohesion by Sulfur Segregation in Ferromagnetic Iron and Nickel —A First-Principles Study—
1.48 J/m
2.69 J/m2
Table 3
2
1.33 J/m
2.29 J/m2
Calculated total magnetic moment.
bcc Fe3(111)[11 0]
fcc Ni5(012)[100]
GB cell
180.1 B /76Fe
55.2 B /84Ni
FS cell
175.2 B /76Fe
54.5 B /84Ni
Bulk cell
257.5 B /120Fe
73.0 B /120Ni
region, it is considered that Poisson’s ratio does not affect
significantly. Since the second derivative at x ¼ is equal
to zero ( f 00 ðÞ ¼ 0), the maximum tensile stress, max , is
equal to
f 0 ðÞ ¼
2
expð1Þ:
This tensile strength, max , is not an accurate but an
approximate value. However, it is convenient and fast way to
obtain max , because this method avoids difficult calculations
in largely strained system and can calculate all points
independently. However, it is necessary to confirm that the
fracture plane that is set at the beginning of the calculation is
reasonable; a few different fracture planes should be tested to
obtain the minimum cohesive energy and tensile strength.
Segregation energy for a S atom, Eseg(site) / eV atom
GB energy
FS energy
fcc Ni5(012)[100]
0.5
0.0
-0.5
-1.0
-1.5
FS
-2.0
GB
-2.5
Inner bulk
-3.0
0
1
2
3
4
5
6
7
Segregation site
8
9
10
11
(b) Ni Σ 5
-1
2
Segregation energy for a S atom, Eseg(site) / eV atom
bcc Fe3(111)[11 0]
(a) Fe Σ 3
-1
Table 2 Calculated GB energy, and free surface (FS) energy that appears
after the GB is fractured.
2685
0.5
Not conv.
0.0
-0.5
-1.0
-1.5
-2.0
-2.5
GB
Inner bulk
FS
-3.0
-3.5
0
1
2
3
4
5
6
7
8
9
10
11
12
Segregation site
3.
Results and Discussion
3.1
Grain boundary energy, free surface energy, and
magnetic moment
Table 2 shows grain boundary (GB) energy and free
surface (FS) energy. It is well known that grain boundary
energy is about 2/3–1/2 of free surface energy. The grain
boundary energies for bcc Fe3 and fcc Ni5 GBs are about
2/3–1/2 of these free surface energies, which is in agreement
with the well-known trend.
Table 3 shows the total magnetic moments for three cells
as shown in Fig. 2. From this table, the average magnetic
moment per Fe(Ni) atom for bulk cell is 2.15(0.61) B /atom,
which is in good agreement with experimental results (Fe:
2.2 B /atom, Ni: 0.6 B /atom).
3.2 Segregation energy
First, we show the segregation energy of one sulfur atom.
Figure 5 shows the calculated segregation energies when one
S atom is introduced into the GB vacancy sites (site 0) or
substituted for a Fe/Ni atom at various sites (site 1–11 or 12).
We can find the following things common to the Fe3 and
Ni5 cases.
In the GB region, the site 2 is the most favorable
segregation site and the site 0 is the second one. The site 1
and 3 are not so favorable comparing with the site 2 and 0.
The GB segregation energies at site 2 and 0 for both Fe and
Ni cases are larger than 1:0 eV/atom, which is very
Fig. 5 Calculated segregation energies for a S atom in (a) bcc Fe3 GB
and (b) fcc Ni5 GB. Roughly speaking, the 0–3 sites are in grain
boundary (GB) region, the 4–8,9 sites in inner bulk region, and the 8,9–
11,12 sites in free surface (FS) region. The energy at site 9 in (b) is not a
converged result.
(negative) large value as can be seen from Fig. 3. Note that
the same argument holds for site 0, 1, 2, and 3, because
the grain boundary we use is a symmetrical one; there is a
mirror plane at the grain boundary plane.
As can be seen from Fig. 1, site 2 and site 2 make a
dumbbell pair, because the distance between the two sites is
very close (about 0.25 nm). This indicates that the environment (structure) around site 2 is very different from the inner
bulk site. This is the reason why the segregation energy of the
site 2 is very large.
In the free surface (FS) region, the outermost site (site 11
for Fe3 and site 12 for Ni5) is the most stable site for S
atom. Its surface segregation energy is negative and larger
(more stable) by 1.3–1.5 eV/atom than any GB segregation
energies. This energy difference between the GB and surface
segregation energies is known as embrittling potency energy
in the Rice-Wang model,10,11) which energy indicates the
extent of decohesion (embrittlement) induced by the impurity
(solute).
From these results we can understand that sulfur favors
grain boundary rather than inner bulk, and furthermore favors
2686
M. Yamaguchi, M. Shiga and H. Kaburaki
0.0
0
1.8
Area density of segregated S atoms / nm-2
3.6
5.4
7.2
9.0
10.8
eV
total
seg /
Total segregation energy, E
0a
2a
-3
2a0d
0a0d
2a2b
-4
2b2c0d
2b0a0d
2a2b0d
2a0a0d
0a0b0d
2a2b2c
-5
-6
-7
14.4
2a2b2c0a0d
2a2b2c2d0d
2a2b0a0c0d
2a2b0a0b0d
2a0a0b0c0d
2a2b2c2d0a0d
2a2b2c0b0c0d
2a2b2c0a0b0d
2a2b0a0b0c0d
-1
-2
12.6
-8
Number of layers: site
1
2
3
2 atomic layers: 0, 2
3 atomic layers: 0, 2, 2
0
1.8
2a2b2c2d0a0b0c0d
4
5
6
7
8
12.6
14.4
Total segregation energy, E
total
seg /
eV
0
2a2c0b0c0d
2a2b2c2d0d
2a2c0a0b0d
2a2b0a0c0d
2a2b0a0b0d
2a0a0b0c0d
-2
0a
2a
2a2b2c2d0a0d
2a2b2c2d0b0d
2a2b2c0a0b0d
2a2b2c0b0c0d
2a2b0a0b0c0d
-4
-6
-8
2a2b0d
2a2c0d
2a0b0d
2a0a0d
2a2b2d
0a0b0c
-10
0.0
0.0
11:51 eV
11:92 eV
4
12.6
14.4
Ni
Fe
-0.4
-0.6
2a2b2c0a0b0c0d
-0.8
2a2b2c2d
-1.0
2a2b
-1.2
2a2b2c
2a2b0a0b0c0d
2a0a0b0c0d
2a
-1.4
-1.6
2a
0a0b0c0d
2a2b2c2d0a0b0c0d
0a0d
-1.8
0a0b0c
-2.0
0
0.0
35
5
6
7
8
Number of segregated S atoms in the unit cell
Fig. 6 Calculated total segregation energies for segregated S atoms in (a)
bcc Fe3 GB and (b) fcc Ni5 GB. The characters (2a, 0b, etc.) indicate
the atomic site configurations for segregated S atoms as shown in Fig. 1.
For example, the lowest energy configuration at 4 S atoms in (a) is
‘‘2a2b2c2d’’.
surface rather than grain boundary. This trend is important
for grain boundary weakening as stated later.
Next, we show the segregation energy when two or more
sulfur atoms segregate to the grain boundary. Figure 6 shows
total segregation energy of the system, which indicates that
how much energy the system obtains when sulfur atoms
move from inner bulk region to grain boundary region.
Owing to the restriction of computer resources, we take only
two sites (0 and 2 sites) for the sulfur segregation.
From Fig. 6, we can understand that the largest (negative)
total segregation energy for each concentration (number of
sulfur atom) increases monotonically with increasing concentration. Using the atomic site notations as shown in
Fig. 1, the most stable configurations for Fe3 case are
2a, 2a2b, 2a2b2c, 2a2b2c2d, 2a0a0b0c0d, 2a2b0a0b0c0d,
2a2b2c0a0b0c0d, and 2a2b2c2d0a0b0c0d. For Ni5 case,
the most stable configurations are 2a, 0a0b, 0a0b0c,
0a0b0c0d, 2a0a0b0c0d, 2a2b0a0b0c0d, 2a2b2c0a0b0c0d,
and 2a2b2c2d0a0b0c0d. These results can be partly understood from the fact that the distance between the crystallographic equivalent sites (the 0–0 and 2–2 distance) is 0.4 nm
for Fe3 and 0.352 nm for Ni5, which are much longer
than the 0–2 distance. For Fe3(Ni5), the nearest neighbor
0–2 distance is 0.256(0.223) nm in no segregation case. Since
Maximum tensile stress, σ max / GPa
3
8:32 eV
10:27 eV
1
2
3
4
5
Number of S atoms
6
7
8
12.6
14.4
(b) Maximum tensile stress (Tensile strength)
2a0b0c0d
2a2b0b0d
2a2b2c2d
2a2c0a0d
2a0a0b0d
0a0b0c0d
1.8
Area density of S atom / nm-2
3.6
5.4
7.2
9.0
10.8
Ni
Fe
Cleavage: Ni(012), Fe(111)
30
Clean GB
25
20
15
10
5
0
0
1
(c) Cohesive energy
0.0
6
1.8
2
3
4
5
Number of S atoms
6
Area density of S atoms / nm-2
3.6
5.4
7.2
9.0
10.8
7
8
12.6
14.4
Ni
Fe
Cleavage: Ni(012), Fe(111)
5
Cohesive energy, 2γ / J m -2
2
8
12
Area density of S atom / nm-2
3.6
5.4
7.2
9.0
10.8
1.8
2a2b2c2d0a0b0d
2a2b2c0a0b0c0d
-12
1
7:29 eV
-0.2
2a2b2c2d0a0b0c0d
0
Ni5
4:74 eV
(a) Average segregation energy
2b2c2d0d
2a2b0a0d
2a2b2c0d
2a2b0c0d
2a0a0b0d
0a0b0c0d
2a2b2c2d
Area density of segregated S atoms / nm-2
3.6
5.4
7.2
9
10.8
2b0d
2c0d
2a2b
2a0d
2a2c
0a0d
Fe3
4
2a2b2c2d0a0b0d
2a2b2c0a0b0c0d
Number of segregated S atoms in the unit cell
(b) Ni Σ 5
Num. of S
1 atomic layer: 0 (or 2)
-9
0
Table 4 Total segregation energy. The one and two atomic layers’ cases
are plotted in Fig. 6.
Average segregation energy, E segav / eV atom -1
(a) Fe Σ 3
Clean GB
4
3
2
1
0
0
1
2
3
4
5
Number of S atoms
6
7
8
Fig. 7 (a) Calculated average segregation energies per one S atom in bcc
Fe3 and fcc Ni5 GBs. Segregated sulfur configuration is indicated by
characters (0a, 2a, etc.). (b) Calculated maximum tensile stress (tensile
strength). (c) Calculated cohesive energies. For (b,c), cleavage energy and
tensile strength are also shown for bcc Fe(111) and Ni(012) crystal planes.
Grain Boundary Decohesion by Sulfur Segregation in Ferromagnetic Iron and Nickel —A First-Principles Study—
(a) Clean Fe GB
(b) S at site 0
Fe
Fe
-8
-7
-5
-9
-6
-10 -10
-4
-4
-2
1
1
0
10
S
4
7
6
10
8
7
8
10
5
9
2
10
11
9
11
11
3
4
5
6
7
8
10
1
0
4
6
7
9
1
7
8
10
9
11
-4
-3
-2
S
2
1
3
5
6
-4
S
0
1
4
4
7
1
3
-7
-6
-5
-3
-2
0
2
3
5
6
9
1
0
4
-5
-3
-2
1
-8
-11
-10
-9
-7
-6
-4
-3
1
-8
-11 -11
-10
-9
-7
-6
-5
-4
2
7
8
10
-8
Ni
-11 -11
-10
-9
-6
(f) S at site 0 and 2
Ni
-11
-2
1
3
4
5
7
9
-4
-4
2
4
8
-3 S
1
6
7
-5
-10
-9
-7
-5
(e) S at site 0
Ni
-7
-6
-2
3
4
5
-4
-8
-7
-7
-3
0
2
-9
-8
-7
(d) Clean Ni GB
Fe
-10 -10
-10
10
(c) S at site 0 and 2
2687
11
10
11
Fig. 8 Valence electron density maps in unit of electron/(0.1 nm)3 (a): Clean Fe3 GB. (b): Fe3 GB with 4S segregation at site 0.
(c): Fe3 GB with 8S segregation at site 0 and 2. (d) Clean Ni5 GB. (e): Ni5 GB with 4S segregation at site 0. (f): Ni5 GB with 8S
segregation at site 0 and 2.
neighboring S atoms repel each other in the GB, S atom
occupy the sites with avoiding the neighboring.
We do not consider activation energy from one configuration to the other one; we consider only total segregating
energy. The final segregation configuration should be
determined by the statistical method like Monte Carlo
simulation. At this stage, we can at least say that the two
atomic layers’ segregation simultaneously at site 0 and 2 has
so large segregation energy that this configuration is
considered to be energetically possible to realize.
For further concentration like three atomic layers’ segregation as shown in Table 4, the energy gain from segregation
becomes small. In particular for Ni5 case, the energy gain
from two to three atomic layers’ segregation is only
0:41 eV. This indicates that the segregation tends to
saturate with increasing segregation concentration.
Figures 7(a)–(c) shows the average segregation energy per
one sulfur atom, calculated cohesive energy and tensile
strength of grain boundary.
From Fig. 7(a), we can see that the average segregation
energy increases for Ni5 in negative value with increasing
sulfur concentration up to one atomic layer concentration
(7.2 atom/nm2 ), whereas decreasing for Fe3 case. Over
7.2 atom/nm2 , the average segregation energy decreases in
negative value for both Fe3 and Ni5. We can not explain
clearly these energy changes, because these changes depend
on a small change of metal atoms’ positions and the
environment around the segregated S atoms. The most
important point of these results is that these average
segregation energies for all concentration are (negative)
larger than about 1:0 eV/atom, which is considered to be
very large segregation energy as can be seen from Fig. 3
(McLean’s curves).
3.3 Tensile strength and cohesive energy
From Fig. 7(b), we can see that the tensile strengths for
both Fe3 and Ni5 clean GBs are almost the same and
about 25% smaller than the cleavage cases for Fe(111) and
Ni(012) crystal planes. With increasing sulfur concentration,
the tensile strength decreases. The decrease for Ni case is
slower than for Fe case up to one atomic layer segregation
(7.2 atom/nm2 ). Then, both tensile strengths for Ni and Fe
cases decrease rapidly over 7.2 atom/nm2 and become close
to zero at two atomic layers’ segregation (14.4 atom/nm2 ).
This change at 7.2 atom/nm2 is due to the fact that
segregated S atoms begin to neighbor and repel each other
in the GBs.
From Fig. 7(c), we can see that the cohesive energies for
clean GBs are about 25% smaller than these cleavage cases.
With increasing sulfur concentration, the cohesive energy
decreases. For the two atomic layers’ concentration
(14.4 atom/nm2 ), the cohesive energies become close to
zero for both Fe and Ni cases. Comparing with tensile
strength, the change at 7.2 atom/nm2 is not so apparent in the
case of cohesive energy. As stated later, we think that this is
due to compensation between the energy loss by S-S
repulsion and the energy gain by surface stabilization energy
of S.
3.4 Charge density
Figure 8 shows the calculated electron density distributions for valence electrons. Figure 8 shows clean GB case
(a,d), one atomic layer’s sulfur segregation case (b,e), and
two atomic layers’ segregation case (c,f) for bcc Fe3 and
Ni5 cases, respectively. We can see that sulfur makes
strong covalent bonds with Fe and Ni; it seems that stronger
bonds are seen for Ni case than Fe case. For the two atomic
layers’ segregation cases (c,f), neighboring sulfur atoms (at
site 0 and 2) can not make bonds; even push apart each other
and therefore make a wide vacuum region in the GB regions.
It seems that the neighboring S atoms at the GBs have a
strong repulsive interaction. This interaction should increase
the total energy of the system; it results in the decrease of
segregation energy (i.e. segregation possibility). However,
the average segregation energy does not decrease significantly after S atoms begin to neighbor and repel each other
over one atomic layer’s segregation as can be seen from
Fig. 7(a). Here, we think that the repulsion between
neighboring S atoms tends to make a fracture surface as
can be seen from Fig. 8(c,f), which brings about a stabiliza-
2688
M. Yamaguchi, M. Shiga and H. Kaburaki
Ni calc.
Fe calc.
Ni exp.
700
20
600
500
15
400
10
300
200
5
100
0
Ultimate tensile strength (Exp.) / MPa
Calculated tensile strength, σ max / GPa
25
(2)
0
0
5
10
15
20
25
30
Intergranular concentration / atomic %
35
Fig. 9 Calculated tensile strength for Ni5 and Fe3 GBs and experimental ultimate tensile strength for fcc Ni. Intergranular concentration
indicates the concentration within 0.5 nm region near the GB plane in both
sides. Experimental data for fcc Ni is from ref. 12.
tion energy because the surface segregation energy is much
larger than the GB segregation energy by about 1.2–1.5 eV/
atom as shown in Fig. 5. In other words, the enegy gain by
stabilization of S atom on the metal surface compensates the
energy loss of S-S repulsion in the grain boundary, and
therefore keeps the average segregation energy large.
3.5 Comparison with experiments
Figure 9 shows the comparison between calculated tensile
strength and experimental ultimate tensile strength12) of fcc
Ni. Intergranular concentration is defined as concentration
within 0.5 nm region from the grain boundary plane in both
sides. Although the order of the tensile strength is very
different between calculation and experiment, both of
strengths reduce by one order of magnitude with increasing
sulfur concentration. The discrepancies in intergranular
concentration are due to many differences between calculation and experiment; calculations are done for only one
symmetrical tilt grain boundary (5 for Ni), whereas
experimental fracture occurs at various kinds of grain
boundaries (random grain boundaries, etc.) associated with
dislocation emitting. Considering these facts, the agreement
between calculations and experiments seems to be reasonable.
(3)
(4)
(5)
(6)
(7)
4.
Summary
For ferromagnetic bcc Fe3(111)[11 0] and fcc
Ni5(012)[100] symmetrical tilt grain boundaries, we
simulate grain boundary decohesion (embrittlement) by
progressively adding sulfur atoms to the boundaries. The
results for ferromagnetic Ni case in this work are almost the
same as our previous results for non-magnetic Ni case,1)
which indicates that the spin-polarization does not affect the
grain boundary decohesion significantly for Ni case. We
found the following things in common with Fe and Ni
systems.
(1) When only one sulfur atom segregate to the grain
boundary region in the unit cell, the segregation energy
depends on the segregation site. We found that the three
sites (site 0, 2, and 2) are very favorable in the grain
boundary region, in which the segregation energy is
negative and larger than 1:0 eV/atom.
When two or more sulfur atoms segregate, they
segregate to the grain boundary with avoiding the
neighboring with each other. Up to one atomic layer’s
segregation (4 atoms in the cell, 7.2 atom/nm2 ), the
segregated sulfur atoms tend to avoid being neighboring. This indicates that the neighboring sulfur atoms
repel each other. Up to two or three atomic layers’
segregation, segregated sulfur atoms neighbor and repel
each other. For this reason, the energy gain by sulfur
segregation gradually decreases with increasing sulfur
segregation concentration; it indicates that segregation
tends to saturate.
The (negative) total segregation energy increases
monotonically up to at least two atomic layers’
segregation (14.4 atom/nm2 ). Strictly speaking, the
segregation concentration should be determined by a
statistical method like Monte Carlo simulation considering all segregation configurations and the activation
energy between these configurations.
The tensile strength and cohesive energy decreases with
increasing sulfur segregation concentration. At the two
atomic layers’ segregation (14.4 atom/nm2 ), the both
of tensile strength and cohesive energy reduce to be
one-tenth of the original values. This is in agreement
with experimental results; the tensile strength of bulk
fcc Ni finally become about one-tenth of its original
strength by introducing sulfur.12)
For more than one atomic layer’s segregation (7.2
atom/nm2 ), the sulfur atoms begin to neighbor each
other. Over this concentration, the decreasing rate in
tensile strength and cohesive energy becomes slightly
larger; this trend is more apparent for Ni than Fe. This is
considered to be due to a repulsive interaction between
neighboring sulfur atoms.
Charge density maps clearly show that the neighboring
sulfur atoms in the grain boundary do not make
covalent bonds with each other; even push apart each
other and make a fracture surface (vacuum region) in
the grain boundary. The sulfur atoms seem to be forced
into non-ideal bonding because the nickel-sulfur bonds
are stronger than the sulfur-sulfur bonds.
Up to two atomic layers’ segregation (14.4 atoms/
nm2 ), the energy loss of the neighboring S-S repulsion
seems to be compensated by the energy gain from the
stabilization energy of S atom on the metal surface,
because the S-S repulsion tends to make a fracture
surface in the grain boundary and the stabilization
energy of S atom on the metal surface is much larger
than that in the grain boundary. This may be the reason
why the average segregation energy does not reduce
significantly by the neighboring S-S repulsion up to at
least two atomic layers’ segregation.
Acknowledgments
We thank Y. Nishiyama in JAEA and J. Kameda in
University of Pennsylvania for helpful discussions.
Grain Boundary Decohesion by Sulfur Segregation in Ferromagnetic Iron and Nickel —A First-Principles Study—
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